I was thinking about this and I think it's either Turkey since the Turks took over the Byzantines or the Greeks since the Byzantine culture was very Greek.
When they conquered Constantinople, the Ottoman rulers appropriated the title "Emperor of Rome," but regardless, no country today can really claim to be a successor to the Roman Empire (Byzantine or otherwise).
The "Byzantine Empire" was not a successor state. It was the Roman Empire. The Eastern Roman Empire evolved over its thousand year life, as would most polities, but was the same entity.